3.1021 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^5} \, dx\)

Optimal. Leaf size=75 \[ -\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \]

[Out]

-(a + b*x^4)^(3/4)/(4*x^4) + (3*b*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(8*a^(1/4))
- (3*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(8*a^(1/4))

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Rubi [A]  time = 0.10817, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^5,x]

[Out]

-(a + b*x^4)^(3/4)/(4*x^4) + (3*b*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(8*a^(1/4))
- (3*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(8*a^(1/4))

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Rubi in Sympy [A]  time = 11.9555, size = 68, normalized size = 0.91 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{4 x^{4}} + \frac{3 b \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**5,x)

[Out]

-(a + b*x**4)**(3/4)/(4*x**4) + 3*b*atan((a + b*x**4)**(1/4)/a**(1/4))/(8*a**(1/
4)) - 3*b*atanh((a + b*x**4)**(1/4)/a**(1/4))/(8*a**(1/4))

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Mathematica [C]  time = 0.0490131, size = 67, normalized size = 0.89 \[ \frac{-3 b x^4 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )-a-b x^4}{4 x^4 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^5,x]

[Out]

(-a - b*x^4 - 3*b*(1 + a/(b*x^4))^(1/4)*x^4*Hypergeometric2F1[1/4, 1/4, 5/4, -(a
/(b*x^4))])/(4*x^4*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^5,x)

[Out]

int((b*x^4+a)^(3/4)/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.461705, size = 234, normalized size = 3.12 \[ -\frac{12 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\left (\frac{b^{4}}{a}\right )^{\frac{3}{4}} a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + \sqrt{\sqrt{b x^{4} + a} b^{6} + \sqrt{\frac{b^{4}}{a}} a b^{4}}}\right ) + 3 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + 27 \, \left (\frac{b^{4}}{a}\right )^{\frac{3}{4}} a\right ) - 3 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} - 27 \, \left (\frac{b^{4}}{a}\right )^{\frac{3}{4}} a\right ) + 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^5,x, algorithm="fricas")

[Out]

-1/16*(12*(b^4/a)^(1/4)*x^4*arctan((b^4/a)^(3/4)*a/((b*x^4 + a)^(1/4)*b^3 + sqrt
(sqrt(b*x^4 + a)*b^6 + sqrt(b^4/a)*a*b^4))) + 3*(b^4/a)^(1/4)*x^4*log(27*(b*x^4
+ a)^(1/4)*b^3 + 27*(b^4/a)^(3/4)*a) - 3*(b^4/a)^(1/4)*x^4*log(27*(b*x^4 + a)^(1
/4)*b^3 - 27*(b^4/a)^(3/4)*a) + 4*(b*x^4 + a)^(3/4))/x^4

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Sympy [A]  time = 6.1321, size = 39, normalized size = 0.52 \[ - \frac{b^{\frac{3}{4}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**5,x)

[Out]

-b**(3/4)*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), a*exp_polar(I*pi)/(b*x**4))/(4*x
*gamma(5/4))

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GIAC/XCAS [A]  time = 0.228602, size = 278, normalized size = 3.71 \[ -\frac{1}{32} \,{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{b x^{4}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^5,x, algorithm="giac")

[Out]

-1/32*(6*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 +
a)^(1/4))/(-a)^(1/4))/a + 6*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)
^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a - 3*sqrt(2)*(-a)^(3/4)*ln(sqrt(2)*(b
*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a + 3*sqrt(2)*(-a)^(3/4
)*ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a + 8*(
b*x^4 + a)^(3/4)/(b*x^4))*b